Trigonometry Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

Hi this is www.educator.com and we are going to try more examples of modified sin waves where we start with the basic equation of sin(x) or cos(x) and the graph of sin(x) or cos(x).0000

We introduce these constant which are going to change some of its attributes and we see what that does to the graph.0013

Remember the equation we are working with in general is (a)sin arcos(bx + c) + d and then from each of those values we figured out these various attributes amplitude, period, phase shift and vertical shift.0021

In this particular equation, the amplitude, remember that is just (a) so you read that as 4, the period is 2pi/b , the b is 2 here, so that is 2pi/2 which is pi.0041

The phase shift, that is the strangest one –c/b, that is (–pi/2)/2 or –pi/4.0066

Finally the vertical shift, that is the easier one is -1 here.0085

Again we will start out with the basic sin way, we will work through introducing these attributes one at a time and see how that moves around and create a new function for us.0097

Let us start out with the basic sin way.0108

There is pi, 2pi, so I’m just going to graph sin(x) to start with.0119

I’m going to graph it flat because I’m looking ahead and noticing that the next step is to increase the amplitude to 4.0130

It is really going to be stretched up, I’m going to keep my scale narrow here.0139

The next step is to introduce the amplitude, we are going to graph 4sin(x).0148

That stretches it up and down by a factor of 4, -4 goes down to -4 goes up to +4, that makes it a lot steeper.0156

That is 4sin(x), we got the amplitude incorporated there, remember it is very important to do these in order, amplitude, period, phase shift, vertical shift.0175

Next step is to introduce the period, right now the period is 2pi and we want the period to be pi.0185

Instead of doing a full cycle in 2 pi units, that is going to do cycle in pi units, that means it shuffles twice as fast.0197

Let me try and draw that.0205

That is pi/2 there, 3pi/2, ok we got the period incorporated.0220

What I really graphed there was 4sin(2x), next we want to incorporate the phase shift.0230

I will be graphing 4sin(2x) + pi/2 and this is starting to get a little crowded here, so I’m going to do this one in red.0241

That means that the phase shift is –pi/4, that means we move –pi/4 units to the left.0257

Instead of starting at 0, I’m going to start at –pi/4 and it came back down before pi/2 so now it is going to come back at pi/4.0267

It used to come back up at pi and now it comes back up at 3pi/4.0287

That red graph is what we get by incorporating the phase shift and finally we will incorporate the vertical shift and I will do this one on blue.0301

We are going to be doing 4sin(2x + pi/2) -1, that means we take the graph and we move it down by one unit.0316

It is not the same as changing the amplitude where things stretch up and down.0328

Here we are not stretching anything, we are just moving everything directly down by one unit.0331

Instead of starting at 0 going up to 4 down to -4, everything is moved down by 1 unit.0336

It will peak at three and instead of going down to -4, it will go down to -5 and the middle part will be at -1.0346

Let me go and try to graph that in blue.0355

It comes back to -1 instead of 0, bottoms out at -5 instead of -4, goes back up peaks at three, back down to -5 and back up to -1 again.0362

This blue curve is the final graph there, it is very complicated by the time it is all over but it is a bunch of simple steps.0382

The first step is to look at the equation and identify these quantities, amplitude, period, phase shift, and vertical shift.0395

We got pretty simple equations for each one of those, the tricky part of the graphing is where you start with sin(x).0403

You remember how to graph sin(x), that is very easy and then start incorporating these attributes one by one.0414

Amplitude stretches it vertically up and down, period stretches or compresses it horizontally, in this case the period was pi and since it was 2pi/4 we have to compress it by a factor of 2.0420

Phase shift moves it over to left and right, vertical shift moves it up or down.0434

We finally end up with this blue curve that has been modified according to all those attributes.0440

Now we are being asked to find a cos wave and we are given the attributes but we are not given the equations.0000

We have to reverse engineer the equation from this attributes.0007

Remember the equation we are going to go for is (a)cos(bx+c)+d, but we do not know what a, b, c, and d are, we got to figure them out from these attributes.0012

Amplitude is the opposite value of a, we will take a=2 to give ourselves amplitude =2.0028

The period is 2pi/b and that is supposed to be equal to 3pi.0048

From that we can figure out if we solve that for b, we get (b) is equal to 2/3, that is how we can figure out what (b) should be.0063

The phase shift is –c/b which is supposed to be equal to pi/2, now that requires a little bit of solving, so we get –c/b we already figure out that was 2/3 is equal to pi/2.0073

That is -3(c)/2 is equal to pi/2, the 2 is cancel and so we get (c) is equal to –pi/3.0106

That is how we figured out what (c) is, we reversed engineer this equation –c/b.0122

Finally the vertical shift, is (d) which is -2, we figured out what a, b, c, and d are, that means that our equation is a=2cos(2/3x)-(pi/3-2).0128

We figured out the first part of the problem which is to find the equation, now we are going to do the tricky part which is graphing it.0162

It is not so tricky if you start with the basic cos wave if you remember how to graph that and then you introduce this attributes in the right order.0168

I will start with the basic cos wave, cos(x), there is pi, 2pi, 3pi. Cos(x) I know it starts at 1, there is 1, -1, it starts at 1, it goes down to 0.0179

Bottoms out at -1 goes back up to 0, goes back up to 1, now I got cos(x) that I pretty much do from memory.0217

Next I’m going to do is introduce the amplitude, I’m going to talk about 2cos(x) that just stretches it vertically up and down, it goes up to 2 and down to -2.0228

Next I’m going to adjust the period, what I’m actually going to be graphing is 2cos(2/3x).0257

I know that makes the period 3pi, our original period is 2pi, this is stretching it out by a factor of 50% instead of doing a full cycle in 2pi, it is going to do a full cycle in 3pi.0269

Let me give my self some coordinates here, it still going to start at 0 but it is going to bottom out now.0286

It is going to finish up at 3pi instead at 2pi, it is going to bottom out halfway between them that is 3pi/2.0295

I took the graph and I stretched it out so that it has a period of 3pi now, phase shift –pi/3.0324

What I’m going to be graphing is 2cos(2/3x)-pi/3, sorry the phase shift is pi/2, the c value is –pi/3 but the phase shift is pi/2.0338

I’m going to take this graph and move it over pi/2 units to the right.0358

It looks like it is getting a little complicated now, I’m going to switch into red here.0364

We are going to graph the phase shift in red so that means instead of starting at 0, I’m going to start at pi/2.0371

It is going to bottom out at 2pi and it is going to come back up at 3pi + pi/2.0379

My orientation points were pi/2, 2pi, 3pi + pi/2, because I just took my orientation points before and move them all over by pi/2.0413

That incorporated the phase shift of pi/2, we are almost done here.0425

We got on more step to incorporate which is this vertical shift of -2.0430

I’m going to do this one in blue so we are going to be graphing 2cos(2/3x)-(pi/3-2) that means we are going to take the whole graph and just move it down 2 units.0435

Now remember, before our peaks and values were 2 and -2, we moved that down by 2 units and the peaks and values are now going to be at 0 and -4.0455

I’m going to extend my axis down to -4, -3, -4.0467

I’m going to take my reference points from before and move them down by 2 units.0476

This graph peaks at 0 and it never actually goes north of the x axis because it is going up to 0 and down to -4.0505

That is the final graph that we have been asked for, let us recap there.0520

We are given all these attributes amplitude, period, phase shift, and vertical shift.0528

We know the equation for those 4 things, absolute value of a, 2pi/b, -c/b and d.0533

What we do is we plug in those attributes and reverse engineer to figure out what a, b, c, and d are.0541

Then we can put those together to get the equation of the function that we are trying to graph.0550

To actually graph it, we start with basic cos curve, we introduced the amplitude.0557

It is very important to do this in order, introduce the amplitude where you expanded it up, up and down.0563

The thing actually stretches out, we introduced the period which collapses it horizontally or stretches it out horizontally.0570

In this case, we are changing the period from 2pi to 3pi, so that stretches it horizontally.0581

We introduced the phase shift which moves it horizontally to the right or left without stretching it.0587

The period was the part where you stretch it, the phase shift just moves it without stretching it.0593

Finally, we introduced the vertical shift which moves it up and down without stretching it, in this case it moved down.0599

At the end of it, it is quite a complicated process but individually each one of these steps if you keep them in order is not to hard.0608

You just start with you original cos graph and then move them around according to these steps until you get the one you want.0615

That is the end of our lecture on modified sin waves, this is www.educator.com.0622

Hi this is educator.com and today we're going to learn about modified sine waves.0000

We're going to learn how to analyze and graph these functions given by, instead of just talking about sin(x) and cos(x), now we're going throw a whole bunch of constants in there.0007

We'll talk about asin(bx+c) and then throw a constant outside +d and the same kind of thing with cosine.0020

That takes a basic sine or cosine graph and it moves it all around.0027

We're going to learn some vocabulary to describe those movements and we're going to learn how to graph those.0030

First of all, some vocabulary, remember we're talking about the graph asin(bx+c)+d.0038

We're talking about sine waves, these are functions that basically have this shape like sin(x), but they maybe moved in different waves.0057

We need some vocabulary to describe the different ways that could be moved.0073

The first one that we're going to learn is amplitude.0079

The amplitude of the sine waves is the vertical distance from the middle of the waves to the peaks.0082

What that represents graphically is this distance right here, that's the amplitude.0089

Of course, that's the same as this distance, that's also the amplitude but you can measure it either way.0102

If the wave is moved up, if it's floating up above, the x axis, somewhere like that, then the amplitude is still the distance from the middle of the wave to the peaks.0114

In terms of equations, it's very easy to spot the amplitude.0129

When you're given asin(bx+c)+d, the amplitude is just that number a, or if the a is negative, you just take the absolute value.0133

It's just the positive version of that number a that tells you the amplitude right there.0148

It tells you how far up away it's going to the peaks, how far down it's going to the valleys.0158

The period of a sine waves, I'll show this in red, is the horizontal distance for the wave to do one complete cycle from one peak to the next peak.0160

That is the period right there, of that wave, that's the period.0179

On that one, that's the period right there.0189

Remember, when you're working out the equations, remember that if you have sin(x) that has period 2π, it takes 2π to repeat itself.0198

We learned that when we looked into the original sine graphs.0212

If you have sin(2x), that makes it wobble up and down twice as fast, the period would be πsin(4x), the period would be π/2.0215

The pattern that you noticed here is it that the period is given by 2π over the coefficient of x, 2π/b.0230

The b there tells you where the period is, not the b itself but you plug b into that equation and that tells what the period is.0242

Two more vocabulary words we need to learn, the phase-shift of a sine wave is the horizontal distance that the wave is shifted from the traditional starting position.0254

Let me rewrite the equation here, asin(bx+c)+d.0269

The traditional starting position for sine would be at (0,0), and the traditional starting position for cosine would be at (0,1).0280

Those are the traditional starting positions but the phase-shift will move the graph to the right or the left.0302

In these equations, it's given by -c/b, and that seems a little mysterious and let me explain that a little bit.0311

We can write this bx+c, first of all, we can factor b out, we can write that b[x + (c/b)].0320

Then, we can write that as b[x - (-c/b)].0332

That's where that -c/b comes from.0340

If you have trouble remembering that formula -c/b, you can go through this little process to remember that, x-(c/b), that shows you that it moves it c/b units to the right.0342

Let me draw that, I'll draw that in blue.0367

If you're starting with a sine curve, the phase-shift is the amount that it moves over, that's the phase-shift right there, it moves it over -c/b units.0370

If you're starting with a cosine curve, that's that phase-shift right there.0393

Finally, the vertical shift is what happens when you take the graph and you just move it up or down vertically without changing anything else.0407

Let me start with a sine curve.0426

If we apply a vertical shift to that, that amount right there, I'll draw this in red, that's the vertical shift.0434

That's a little bit easier to pick out than some of the others, because that's just the d in the original equation.0455

If you're moving the graph up or down by an amount of d.0464

This can get pretty tricky we're starting with the basic sine and cosine curves, but then we're moving around and stretching them out, and moving them up and down in all different ways.0469

It's a little bit tricky but we'll go through some of the examples and you'll get the hang of it.0479

In our first example here, we're given an equation 3cos(4x+π)+2.0485

We have to identify all these various things, the amplitude, the period, the phase-shift and the vertical shift and then we're going to draw a graph of the function.0495

The key thing here is it if you identify these things in order, then it becomes very easy to pick them out using the equations.0502

The graph isn't too hard as long as you these things in order.0510

The amplitude, remember, that's just the number on the outside, the a in the original equation.0517

Let me write down the original equation, acos(bx+c)+d.0528

The amplitude is just the a right there, that's the 3.0539

The period is 2π/b, the b there is 4, that's 2π/4 which is π/2.0543

The phase-shift is -c/b, our c here is π, -π, b is 4, that's -π/4.0563

The vertical shift is just that last term d, which is 2.0580

Those are the answers to the first part of the question.0596

Trickier part is doing the graph and there's a general strategy for doing this graphs that sort of always works, but you really have to follow it closely.0602

The strategy is to start with the basic cosine graph, which hopefully you remember how to do it, you start with the basic cosine graph.0611

Then move it around according to each one of these parameters.0620

The key thing here is you have to do it in order, you have to do amplitude period phase-shift then vertical shift.0624

Let's see how that works out.0630

Let me draw a basic cosine graph and then we'll try moving it around according to these different parameters.0634

Remember, basic cosine graph, there's π, there's 2π, π/2, 3π/2.0645

Basic cosine graph starts at 1, goes down to 0 at π/2, -1 at π, up to 0 at 3π/2, and it's 1 again at 2π.0661

That's the basic cosine graph, you pretty much have to remember that to get started here.0676

First thing we're going to do, is we're going to change the amplitude.0682

Let me keep track of this as I go along.0688

First one we graphed was cos(x), just y=cos(x).0692

Next we're going to graph is 3cos(x), we're going to bring in the amplitude.0697

What that does is that it stretches up the peak, then it stretches down the valleys by a factor of 3.0700

I'm going to draw the same shift graph but three times as tall, and three times as deep.0709

Instead of going from 1 to -1, goes down to -3 and up to 3.0720

That second graph I drew there was 3cos(x).0725

The next one is to introduce is the period.0734

The period is supposed to be π/2, remember, the period is the amount of horizontal distance between one peak and the next peak.0738

My current peak is...0747

I'm going to change the period to π/2.0754

What I'm really graphing here is 3cos4x.0760

What that's going to do is, instead of having a period 2π, it shrinks it horizontally, or it compresses it horizontally so that it does a complete period in the space of π/2.0769

There's π/2 right there.0784

I need to do a whole period between there and there.0787

Every π/2, it does a complete cycle.0800

What I just drew was 3cos4x.0811

The phase-shift I'm going to introduce is -π/4, that means it moves it π/4 to the left and it's getting a little bit messy.0817

I'm going to see if I can draw this in red or will see if it's still visible.0827

Instead of going from 0 to π/2, I'm going to draw my graph from -π/4 to π/4, because we're moving it to the left by π/4.0832

That's the graph in red there, 3cos4x+π because I've introduced the phase-shift in there.0852

Now, it's really going to get messy if I try to draw any more on the same axis, so I'm going to set up a new set of axis.0860

We had π/2, π, -π/2.0876

I'll redraw the red one on this set of axis, -π/2, and that one is going from 3 to -3.0883

Remember the red one is the previous graph shifted over by -π/4.0911

Finally, we need to introduce the vertical shift of 2, and I'll do this last graph in blue.0916

That takes the entire graph and raises it up by 2 units.0922

That means instead of going to 3 from -3, it's going to go up in π, and instead of going down to -3, it only goes down to -1.0927

Let me label that more clearly, -1.0942

I'll draw this one in blue, this is now 3cos4x+π, and I'm introducing the vertical shift of +2.0947

I'm taking the graph and I'm moving it up 2 units there.0965

That blue graph at the end is our final function.0972

This is really a pretty complicated process.0980

There's a lot of steps involved but each individual step is not that hard, and if you do them in order and you're careful about each one, it's not too bad.0983

Let me just recap there, we started with the original graph of cos(x), that's the starting point.0993

Then we introduced the amplitude, and that stretches it vertically by a factor of 3, stretches it up and down.1000

We introduced the period which compresses it horizontally.1013

We introduced the phase-shift, which takes the whole thing and it moves it to the right or the left.1020

Finally, We introduced the vertical shift which takes the whole thing and moves it up or down.1030

Remember, it's important to do these in order, amplitude, period, phase-shift, vertical shift.1041

If you do those out of order, then they'll mess each other up as you go along.1050

We really want to do those in order.1055

We want to practice several of these, so let's get moving in other example.1059

Here's another example, same questions here, amplitude, period, phase-shift, and vertical shift of the following function.1061

Remember we can read this off quickly, just remembering the formula asin(bx+c)+d.1070

If we can figure out what a, b, c, and d are, we have formulas for all of these properties.1080

Amplitude, that's the a, or if the a is negative, make it positive, that's the absolute value of a, which is just 2 here.1089

The period is 2π/b, the b is 2 here, so that's π.1098

The phase-shift is -c/b, which is, okay, c is -π/3, so negative of that is (π/3)/2 will give us π/6.1109

Finally, the vertical shift is d, which ,in this case, is just 0.1135

Finally, the fun part, we get to graph the function1154

Remember, you always start with your basic sine or cosine graph and then you start moving it around according to these different parameters but you got to keep these parameters in order.1159

Let me start with this one's a sine graph.1169

I know the basic shape of the sine graph, I've got that memorized, π and 2π.1174

I know sine always starts at 0, goes up to 1, comes back down to 0, to -1, and then back to 0.1187

There's 1, -1, there's π, π/2, 3π/2, so that's my basic sine graph.1199

That's the first thing I graphed there, sin(x), remember this is the graph of sine and cosine x.1209

Now, we start introducing these other attributes and it's important to go in order.1217

First of all, we're going to introduce the amplitude.1222

The amplitude is 2, but we're really multiplying the graph by -2, -2sin(x) is what I'm going to graph next.1225

-2sin(x) that stretches it vertically because of the 2, but it also flips it vertically.1235

Instead of starting by going up, it's going to go down, goes down to -2, up to 0, up to 2, and back down to 0.1246

That's first one was a little lop-sided, let me just see if I can make that a little more, a little smoother.1270

Okay, we've got -2sin(x), look it's got a bigger amplitude than the original graph and it's flipped over because of the negative sign.1280

Next thing I'll introduce is the period.1288

The period is supposed to be π, so I'm graphing -2sin(2x).1292

That speeds the whole thing up, it shortens the period because the period is now π instead of 2π.1298

I need to do that entire graph in the space of π instead of 2π.1306

There, that one that I just graphed, I shortened the period to be π instead of 2π.1330

The period is now π on that new graph.1337

Next, we're going to do the phase-shift, that's π/6 units to the right.1340

The phase-shift, what I'm about to graph is -2sin(2x)-π/3, so that takes the whole graph and it shifted over π/3 units to the right.1349

Let me do this one in red.1364

I'm going to take that last graph and shift it over π/3 units to the right.1370

Instead of starting at (0,0), it starts at π/3, it's going to come back down to 0, at 5π/6.1374

The phase-shift is supposed to be π/6, so I'm going to move everything over by π/6.1389

Instead of starting at 0, I'm going to start at π/6 and come back at (π/2)+(π/6) which is actually 2π/3.1408

That red curve that I graphed there is -2sin(2x)-π/3.1439

The last step is to do the vertical shift which is 0, so we don't have to move the graph at all which means we're done.1453

This last graph is the one we want.1464

Again, it's a matter of breaking these equations down into their parts.1470

It's very complicated if you kind of look at the whole thing but if you look at each steps and you keep each steps in order then it's not to hard.1475

Remember the equations for amplitude, period, phase-shift, and vertical shift.1484

Once you've got those, you start with your basic sin(x) graph then you change the amplitude which stretches it out up and down, or might flip it.1489

You change the period which compreseses it horizontally.1500

You do the phase-shift which takes the whole thing and without compressing it, it moves it to the right or the left.1507

Finally, the vertical shift moves it up or down.1511

You could just keep moving these graphs around until you build up the equation you're looking for.1515

This one's a little bit different, we're asked to find the sine wave.1526

This time we're told what all the properties are.1530

We're given the amplitude, the period, the phase-shift, and the vertical shift.1535

We want to find an equation and we want to graph the function.1540

I'm going to kind of build this up, the same way we're building the earlier graphs.1543

I'm going to start with the basic sin(x), so that's my basic sine wave.1548

Now, I'm going to give an amplitude 2, and remember 2 is just a number on the outside, the a.1554

Let me rewrite that equation, asin(bx+c)+d.1559

Amplitude 2 means a is 2, 2sin(x).1566

Now, period 4π, remember our equation for period was 2π/b, that is supposed to be equal to 4π.1575

When you solve that out, that tells us that b=1/2, so the b has to be equal to 1/2.1597

That means our equation is now 2sin(1/2)x, so we've incorporated the period.1610

Phase-shift is supposed to be π/2, but remember the phase-shift, our formula for that is -c/b.1622

b is already, we figured out as 1/2, so that's -c/(1/2).1643

If we do a little bit of algebra here, we get (1/2)π=2c, so c=-π/4.1650

I got that from the equation -c/b=π/2.1667

I already figured out my b, now I can figure out my c.1672

The next part of that equation is 2sin[(-1/2)x-(π/4)].1677

Finally, I want to talk about the vertical shift, which is supposed to be 1 and that's the d.1692

Finally, my equation is 2sin[(1/2)x-(π/4)]+1.1704

That is the sine wave that I'm looking for.1715

Now, I want to graph that thing.1722

I start out with the basic sine wave, π, 2π, π/2, 3π/2.1730

Basic sine wave starts at 0, goes up to 1, back down to 0 at π, down to -1, back up to 0 at 2π, basic sine wave.1742

Now, I'll introduce these properties in order.1753

I'll start out with the amplitude.1755

Amplitude's supposed to be 2, so I'm going to stretch this thing up instead of going from 1 to -1, it's going to go up from 2 up to 2, and down to -2.1760

It'll stretch the thing up, and down.1774

Period is 4π, that means the thing stretches out, so that it'll only does one cycle every 4π.1783

That means I have to extend my graph quite a bit here, 3π, 4π, so I'll stretch the thing out, so it'll only does the cycle every 4π.1794

Now, phase-shift π/2, that means the thing is going to shift π/2 units to the right.1817

I better draw a new set of axis here.1824

I've got π, 2π, 3π and 4π, 5π.1835

What I want to do is take the graph I have above and move it π/2 units to the right because we have phase-shift π/2.1850

Instead of starting at 0, I'm going to start at π/2, I go up to there, back down to 0 at 2π plus π/2, back down to -2 between 3 and 4π and back up to 0 there.1866

So, connect these up.1887

Finally, I have to do one with vertical shift 1, that means I'll take the whole graph and I'll move it up by one unit.1913

That means instead of going or peak at 2, it's going to peak at 3 now.1920

Instead of going down to -2, it gets moved up by 1 unit so it's going to go down to -1.1932

Let me draw this last final curve in red.1940

Everything gets moved up by 1 unit.1948

I'm going to plot some points here, moving everything up by 1 unit.1953

That final curve is the one we want.1976

That's 2sin[(1/2)x-(π/4)]+1.1980

Again, it's a complicated procedure but if you take it step by step, each one of the steps is not too hard.1991

First, we kind of reconstructed the equation from these parameters that we were given.1997

Basically we figured out a, b, c and d from these parameters that we were given by sort of reverse engineering the formulas 2π/b, -c/b, and the d.2004

Then, we went through step by step.2020

We started with the basic sine curve.2024

We changed its amplitude, stretched it vertically.2030

We changed its period which stretches out horizontally.2034

We changed its phase shift which moved it over horizontally not stretching but just moving it without stretching.2036

Then, we did the vertical shift, moving it up or down.2041

You should practice a few of these curves on your own.2045

We'll come back and try some more examples together later.2047